Shamsur's 96 rocketed the home side to 309-6, steaming past New Zealand's 307 with four balls and four wickets to spare at the packed Khan Shaheb Osman Ali stadium at Fatullah just outside Dhaka.
The 25-year-old opener hit seven fours and four sixes in his 107-ball knock as Bangladesh successfully chased down a 300-plus total for only the second time in their playing history.
Shamsur, playing only his second one-day international, fell short of what would have been a well- deserved century when his attempted drive off a Corey Anderson delivery became an edge and flew to wicketkeeper Luke Ronchi.
Nasir Hossain (44 not out) stayed till the end to ensure Bangladesh never lost their way in the chase on the batting-friendly pitch.
Earlier, Ross Taylor hit his eighth one-day century as New Zealand posted their highest total in the series.
The visitors were desperate to avoid another humiliating whitewash, after their previous one-day series against Bangladesh.
Taylor, who was part of the New Zealand squad that lost 4-0 to Bangladesh in 2010, remained unbeaten on 107 while Colin Munro made 85 to give the tourists a fighting chance of victory.
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